I study the corrections engendered by the insertion of a “melon” graph in the bulk of the first-order spinfoam used for the graviton propagator. I find that these corrections are highly nontrivial and, in particular, that they concern those terms which disappear in the Bojowald-Bianchi-Magliaro-Perini limit of vanishing Barbero-Immirzi parameter at fixed area ($\gamma \to 0,\gamma j\sim cst$). This fact is the first realization of the often cited idea that the spinfoam amplitude receives higher order corrections under the refinement of the underlying two-complex.

• Received 9 October 2013

DOI:https://doi.org/10.1103/PhysRevD.89.064021